You think that non-rhyme is hard?
Try “how now brown cow” in Northern Ireland. The first two rhyme but not the others.
To me it sounds like “high nigh broyn cuay”.
Unless I make an effort, “get” comes out more like “git.” I can see what you mean about “thing” and “sing.” They rhyme for me now, but when I was a kid, before I started trying to eradicate my accent, “thing” was more like “thaing,” and “finger” was “fainger.” I can’t think of any other words that followed that pattern, though.
I am from Louisville, Kentucky, where the prevailing accent is a blend of Midwest and hillbilly. My dad is originally from St. Louis. We probably have similar accents.
“Wet” and “Get” rhyme enough. (I tend toward “git” more than “get”) not entirely, but enough to make Dr. Seuss work and I don’t have to stretch too far to make them rhyme.
“Thing” and “Sing” absolutely rhyme.
I’m from the Pacific Northwest.
The first and third, which kind of converge for me. BTW, while I can sort of, kind of hear the difference. If I heard a tape of someone from somewhere where they don’t merge saying “Mary” and “Merry” the words do not sound the same. However, I don’t assign any importance to the difference I hear. To me, your M"eh"ry, M"ay"ry and M"a"ry are all alternate ways of saying Mary/Merry/Marry and none of those ways is more standard than the other for any of those words. It’s the context of the sentence that lets me know which word you’re saying, not the vowel sound.
I’m gonna be in NI on Friday and will check this. 
Yup, this.
California
Thing and sing rhyme.
Wet and get rhyme.
Mary, merry, and marry are homonyms.
Both pairs rhyme. Arizona.
Here are various audio samples - warning close-up Irish teeth and acne.
Well now I’m confused. In the audio you linked some people do indeed say how now, and cow as rhyming, but some of them (specifically the Belfast accents I think) pronounce now differently to how or cow. Brown doesn’t rhyme with the other three in any dialect I know.
Sung? 
They rhyme perfectly in South African English.
You don’t want to hear those words in my New Zealand accent. Your skin will crawl.
Even amongst people that say thang, I also hear them say sang. It’s because ing is always pronounced that way. I had no idea some people said thang but would still say seeng (or possibly sihng).
.
Sorange.
Both pairs rhyme, definitely. I’m from NYC.
Both pairs rhyme. Ohio.
Around here, all nine words rhyme perfectly (they all rhyme with “merry”). When I moved to NYC I had to learn how to pronounce these words differently, so natives could understand me. Then, when I moved back here 25 years later, I had to re-learn to pronounce them all the same.
Do people really say “John Carpentar’s The Thang”?
That’s just all kinds of wrong.
I grew up in New Jersey, with parents from Michigan. Both pairs of words rhyme to me, and would to them.
I used to teach in Pennsylvania, and one day I spoke to the kids about different pronunciations.
They pronounced “creek” to rhyme with “thick”, while I pronounce it to rhyme with “seek”. There were other examples.
Same here, only it’s just been the past 3 years for me.
That’s still more than enough to have learned it by heart. Especially after my son’s hopped on Pop 10,000 or so times. (“STOP! You must not hop on Pop” deflects him only briefly, needless to say.)